While preparing a tutorial session for my students, I come across a very challenging integration, which is :$$ \int_{-\infty}^\infty e^{t^2 }\ \mathrm{d}t$$

I attempted many methods to solve it but with no use. Your assistance will be greatly appreciated.

  • $\begingroup$ Can you do $\int t e^{t^2} dt$. $\endgroup$ – Z Ahmed Jan 16 at 16:01
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    $\begingroup$ Depends on how you interpret $\int$. If it means $\int_{-\infty}^\infty$, then this is a well-known result. If you looking for an antiderivative, then your tutorial session is not good... $\endgroup$ – user9464 Jan 16 at 16:05
  • $\begingroup$ Thank you for our response . well it is integration from - infinity to + infinity ! what is the well-know result you referred to ? by the way it is tutorial on linear system analysis $\endgroup$ – Safa Jan 16 at 16:11
  • $\begingroup$ Then you should add those bounds in your integral. See en.wikipedia.org/wiki/… $\endgroup$ – user9464 Jan 16 at 16:35
  • $\begingroup$ Great ! This is what I was looking for , thanks alot $\endgroup$ – Safa Jan 16 at 17:15

Assuming you are looking for the antiderivative of $e^{t^{2}}$: $$\int e^{t^2 }\ \mathrm{d}t = \frac{\sqrt{\pi}}{2}\operatorname{erfi}(x),$$where $\operatorname{erfi}(x)$ is the imaginary error function defined as $$\operatorname{erfi}(x) = \frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{t^{2}}\mathrm{d}t.$$ There is no way of expressing the antiderivative of $e^{t^{2}}$ in terms of elementary functions.

  • $\begingroup$ Brilliant !!! wow you are a calculus genius . Thank you so much . $\endgroup$ – Safa Jan 16 at 16:17
  • $\begingroup$ From where can learn such advance calculus ? I should be grateful if you would recommend a resource to me . $\endgroup$ – Safa Jan 16 at 16:18
  • $\begingroup$ Honestly the wikipedia page is a pretty good start if you just want some of the terminology. Error functions are just one of many types of special functions. $\endgroup$ – DMcMor Jan 16 at 16:20
  • $\begingroup$ luckily I found a video on YouTube that solves the same problem .But he uses erf(x) function instead ! youtube.com/watch?v=jkytxdedxhU $\endgroup$ – Safa Jan 16 at 16:47
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    $\begingroup$ Well noted . many thanks $\endgroup$ – Safa Jan 16 at 19:46

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